Stochastic differential equations and applications

Abstract

[en] In this paper, how to obtain stochastic differential equations by using Itô Stochastic integrals is treated. We will refer to stochastic differential equations as SDE. Then, the theory inderlying the Itô calculus is carefully studied and a thorough analysis of the relationship of the class of processes $M^{2}$ and the space of integrable functions $L^{2}$ is considered. Moreover, under which assumptions a solution of a SDE exists and is unique is provided. Some particular cases of Itô stochastic integrals and SDE are guaranteed throughout a sequence of examples that are linked up with the abstract theory. Finally, the basic ideas and techniques underpinning the simulation of stochastic differential equations are shown. In particular, the Euler-Maruyama method is presented and suitable simulation scenarios are derived from the SDE models developed.